Landau levels and different magnetic-field-driven dynamics of electrons in 2D materials

We studied an electron in two-dimensional lattices subjected to a constant magnetic field. Both square and honeycomb lattices were explored to examine the dynamic properties of the particle. By employing lattice discretization and the magnetic-field influence, Schrödinger equations were formulated for both lattices. Through the time evolution of a particle initially at rest, we identified distinct regimes characterized by breathing dynamics or stationary behavior of the wave packet. Systematic evaluations involving different magnetic field strengths and initial wave packets demonstrated a correspondence between these steady-state regimes and the eigenfunctions of the lowest Landau levels. Furthermore, the imposition of an initial momentum on the particle resulted in cyclotron dynamics, with measurements of orbital radius and frequency in full agreement with the semi-classical approach. Such dynamics give way to nearly regular oscillations or more complex dynamics as we significantly deviate from the Γ point. Additionally, dynamical regimes characterized by vortices and interference patterns are identified, revealing unique features of the lattices.